In typical differential/Riemannian geometry books, we start with a smooth manifold $M$ and typically introduce the following objects:
- a coordinate chart $(U,x=(x^1,\dots, x^n))$, i.e each $x^i:U\to\Bbb{R}$ is a smooth function.
- This coordinate chart induces a basis of vector fields on $U$, denoted $\left\{\frac{\partial}{\partial x^i}\right\}_{i=1}^n$. This may be abbreviated to $\partial_i$ in some cases. THis is fine if you're only dealing with a single coordinate chart, but if there are two that you're talking about simultaneously, it's better to be more explicit.
- A basis of covector fields $\{dx^i\}_{i=1}^n$. These are the exterior derivatives of the functions $x^i$, and they are the dual to the basis above. Alternatively, you can define a-priori $\{dx^i\}_{i=1}^n$ to be the symbolic expression for the dual basis, and then prove that they are indeed the exterior derivative of the coordinate functions $x^i$.
Next, if we suppose that we have a Riemannian metric $g$ (this is the classical notation for the metric tensor). Then, we introduce the notation
$g_{ij}=g\left(\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j}\right)$, i.e take the inner product with respect to $g$ of the coordinate-induced basis vector fields.
We can store the above functions $g_{ij}:U\to\Bbb{R}$ in a matrix $[g_{ij}]$. This is an invertible matrix, and its inverse matrix is denoted by $[g^{ij}]$, so $g^{ij}$ refers to the $(i,j)$-entry of the inverse matrix to $[g_{ij}]$. A more abstract definition of $g^{ij}$ is as follows. Since $g$ is a metric tensor, in particular it means that the mapping $g^{\flat}:TM\to T^*M$, $g^{\flat}(v)=g(v,\cdot)$ is a vector-bundle isomorphism; its inverse is denoted $g^{\sharp}:T^*M\to TM$. Thus, it allows us to "transfer" the $(0,2)$ tensor field $g$ from $TM$ to $T^*M$, i.e we get a $(2,0)$ tensor field $\tilde{g}(\cdot,\cdot)= g(g^{\sharp}(\cdot),g^{\sharp}(\cdot))$. Then, the components $\tilde{g}(dx^i,dx^j)$ are precisely the functions $g^{ij}$ I defined above.
Next, on a Riemannian manifold, we can talk about covariant derivatives with respect to the Levi-Civita connection $\nabla$. With this, we introduce the Christoffel symbols via the equation $\nabla_{\frac{\partial}{\partial x^i}}\left(\frac{\partial}{\partial x^j}\right)= \Gamma^{k}_{ij}\frac{\partial}{\partial x^k}$, or more explicitly, $\Gamma^k_{ij}=dx^k\left(\nabla_{\frac{\partial}{\partial x^i}}\left(\frac{\partial}{\partial x^j}\right)\right)$.
Now, I typically reserve the notation $\mathbf{e}_i$ to mean the normalized version of $\frac{\partial}{\partial x^i}$, so
\begin{align}
\mathbf{e}_i:=\frac{\frac{\partial}{\partial x^i}}{\left\|\frac{\partial}{\partial x^i}\right\|}=
\frac{\frac{\partial}{\partial x^i}}{\sqrt{g\left(\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^i}\right)}}=\frac{1}{\sqrt{g_{ii}}}\frac{\partial}{\partial x^i}.
\end{align}
Sometimes, I might use $\{\mathbf{e}_i\}_{i=1}^n$ to just mean an arbitrary local basis of vector fields (possibly orthonormal, but not necessarily; another term for this is a moving (tangent) frame). Actually, I personally would drop the boldface, and just say "let $\{e_i\}_{i=1}^n$ be a local basis of vector fields for the tangent bundle", or I also like the letter $\xi$, so I might say "let $\{\xi_i\}_{i=1}^n$ be a local frame for $TM$". However, some people may use $\mathbf{e}_i$ synonymously with $\frac{\partial}{\partial x^i}$, so always read the definition provided by the author first, and decide the meaning based on context.
Now I've only taken a cursory look at Pavel Grinfeld's notation, and here's what I gather (I'll try to match up each point above with a cooresponding one below). Also, he seems to work only in $\Bbb{R}^n$, so there's always a Riemannian metric present, and he seems to like the letter $Z$ alot:
- ' He uses $(Z^1,\dots, Z^n)$ to denote coordinates instead of $(x^1,\dots, x^n)$.
- ' He uses the notation $\mathbf{Z}_i$ for what I've called $\frac{\partial}{\partial x^i}$ above.
- ' I haven't seen him introduce an analogue to $dx^i$, but if he did, he'd just denote it as $dZ^i$.
- ' He writes $Z_{ij}$ in place of $g_{ij}$.
- ' He writes $Z^{ij}$ in place of $g^{ij}$.
- ' He writes $\frac{\partial \mathbf{Z}_i}{\partial Z^j}=\Gamma^k_{ij}\mathbf{Z}_k$ for the definition of the Christoffel symbols. Note this only makes sense because he's working in $\Bbb{R}^n$, which is why we can take partial derivatives of vector fields.
Lastly, he introduces a "contravariant basis" $\mathbf{Z}^i$, defined as $\mathbf{Z}^i= g^{ij}\mathbf{Z}_j$. In the usual notation, it would be written as $g^{ij}\frac{\partial}{\partial x^j}$. In my opinion, this is a confusing step because it is artifically forcing one to work within the tangent space and avoid venturing out into the dual space, which is a perfectly natural place to go. I suggest taking a look at this answer of mine where I give more details.
Perhaps one final thing I should mention is why we use the partial differential notation $\frac{\partial}{\partial x^i}(p)$ to mean an element of the tangent space $T_pM$. One way of viewing it is that the definition of a tangent vector is simply as a derivation (a linear map acting on smooth functions, which satisfies the product rule). This is an abstract definition, but it's technically pretty smooth-sailing. A slightly more geometric way of looking at things is by looking at (equivalence classes of) smooth curves passing through the point $p$. If you're given a coordinate chart $(U,x)$ on the manifold about $p$, and if we let $\{e_i\}_{i=1}^n$ denote the standard basis on $\Bbb{R}^n$, then one example of a curve is
\begin{align}
\gamma_i(t)=x^{-1}(x(p)+te_i).
\end{align}
In words, we have a point $p$ in the manifold, and we can look at it's coordinate representation $x(p)\in\Bbb{R}^n$. We can no move along the $i^{th}$-direction, which yields the line $t\mapsto x(p)+te_i$, and finally we can use $x^{-1}$ to map this back to the manifold. So, the curve $\gamma_i$ is like "moving in the manifold while only varying $x^i$". Now, given this curve, there is a natural way in which we can allow it to "act" on smooth functions, namely for any smooth function $f:M\to\Bbb{R}$, consider $(f\circ\gamma_i)'(0)$, but this is just
\begin{align}
(f\circ\gamma_i)'(0)=\frac{d}{dt}\bigg|_{t=0}f(x^{-1}(x(p)+te_i))=
D(f\circ x^{-1})_{x(p)}(e_i)=D_i(f\circ x^{-1})(x(p)),
\end{align}
where $D_i$ means partial derivative with respect to the $i^{th}$ variable. The last term is what we simply write as $\frac{\partial f}{\partial x^i}(p)$. Because of $f$ being arbitrary, it makes sense to 'identify' the curve $\gamma_i$ with the operator $\frac{\partial}{\partial x^i}(p)$, thereby obtaining an identification between the geometric definition of tangent vectors via curves, and the algebraic version via derivations.
